Abstract
In this paper, we study a stochastic SIS epidemic model with nonlinear incidence rate. By employing the Markov semigroups theory, we verify that the reproduction number R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α) can be used to govern the threshold dynamics of the studied system. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)>1, we show that there is a unique stable stationary distribution and the densities of the distributions of the solutions can converge in L1 to an invariant density. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)<1, under mild extra conditions, we establish sufficient conditions for extinction of the epidemic. Our results show that larger white noise can lead to the extinction of the epidemic while smaller white noise can ensure the existence of a stable stationary distribution which leads to the stochastic persistence of the epidemic.
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